Least-squares inverse filtering

A well-behaved input wavelet, such as (1, -  1/2) as opposed to (-  1/2, 1), has a z-transform whose inverse can be represented by a convergent series. Then the inverse filtering described above yields a good approximation to a zero-lag spike output (1, 0, 0). Can we do even better than that?

Formulate the following problem: Given the input wavelet (1, -  1/2), find a two-term filter (a, b) such that the error between the actual output and the desired output (1, 0, 0) is minimum in the least-squares sense.

Compute the actual output by convolving the filter (a, b) with the input wavelet (1, -  1/2) (Table 2-7). The cumulative energy of the error L is defined as the sum of the squares of the differences between the coefficients of the actual and desired outputs:

${L=(a-1)^{2}+(b-{\frac {a}{2}})^{2}+(-{\frac {b}{2}})^{2}}.$

(14)

The task is to find coefficients (a, b) so that L takes its minimum value. This requires variation of L with respect to the coefficients (a, b) to vanish (Section B.5). By simplifying equation (14), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we get

${{\frac {5}{2}}a-b=2},$

(15a)

and

$-a+{{\frac {5}{2}}b=0}.$

(15b)

We have two equations and two unknowns; namely, the filter coefficients (a, b). The so-called normal set of equations (15a) and (15b) can be put into the following convenient matrix form

${\begin{pmatrix}{5}/{2}&-1\\-1&{5}/{2}\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}2\\0\end{pmatrix}}.$

(16)

By solving for the filter coefficients, we obtain (a, b): (0.95, 0.38). Design and application of this least-squares inverse filter are summarized in Table 2-7.

To quantify the spikiness of this result and compare it with the result from the inverse filter in Table 2-3, compute the energy of the errors made in both (Table 2-8). Note that the least-squares filter yields less error when trying to convert the input wavelet (1, -  1/2) to a spike at zero lag (1, 0, 0).

Table 2-3. Design and application of the truncated inverse filter (1, 1/2) with the input wavelet (1, -  1/2).

Filter Design
Input Wavelet					$w\left(t\right):\left(1,\ -{\frac {1}{2}}\right)$
The z-Transform					$W\left(z\right)=1-\ {\frac {1}{2}}z$
The Inverse					$F\left(z\right)=1\ +\ {\frac {1}{2}}z\ +\ {\frac {1}{4}}z^{2}\ +\ \cdots$
The Inverse Filter					$f\left(t\right):\left(1,\ {\frac {1}{2}},\ {\frac {1}{4}},\ \cdots \right)$
Filter Application
Truncated Inverse Filter						$\left(1,\ {\frac {1}{2}}\right)$
Input Wavelet						$\left(1,\ -{\frac {1}{2}}\right)$
Actual Output						$\left(1,\ 0,\ -{\frac {1}{4}}\right)$
Desired Output						(1, 0, 0)
Convolution Table:
		1	-  1/2			Output
	1/2	1				1
		1/2	1			0
			1/2	1		-  1/4

We now examine the performance of the least-squares filter with the input wavelet (-  1/2, 1). Note that the inverse filter produced unstable results for this wavelet (Table 2-5). We want to find a two-term filter (a, b) that, when convolved with the input wavelet (-  1/2, 1), yields an estimate of the desired spike output (1, 0, 0) (Table 2-9). As before, the least-squares error between the actual output and the desired output should be minimal.

Table 2-5. Design and application of the truncated inverse filter (−2, −4) with input wavelet (-  1/2, 1).

Filter Design
Input Wavelet					$w\left(t\right):\left(-{\frac {1}{2}},\ 1\right)$
The z-Transform					$W\left(z\right)=-{\frac {1}{2}}+z$
The Inverse					$F\left(z\right)=-2\ -\ 4z\ -\ 8z^{2}\ -\ \cdots$
The Inverse Filter					f(t): (−2, −4, −8, …)
Filter Application
Truncated Inverse Filter						(−2, −4)
Input Wavelet						$\left(-{\frac {1}{2}},\ 1\right)$
Actual Output						(1, 0, −4)
Desired Output						(1, 0, 0)
Convolution Table:
		-  1/2	1			Output
	−4	−2				1
		−4	−2			0
			−4	−2		−4

The cumulative energy of the error is given by

${L=(-{\frac {a}{2}}-1)^{2}+(-{\frac {b}{2}}+a)^{2}+b^{2}}.$

(17)

Table 2-7. Design and application of a two-term least-squares inverse filter (a, b).

Filter Design
Convolution of the filter (a, b) with input wavelet (1, -  1/2):
	1	-  1/2		Actual Output	Desired Output
b	a			a	1
	b	a		b − a/2	0
		b	a	−b/2	0
Filter Application
Least-Squares Filter				(0.95, 0.38)
Input Wavelet				(1, −0.5)
Actual Output				(0.95, −0.09, −0.19)
Desired Output				(1, 0, 0)

Table 2-8. Error in two-term inverse and least-squares filtering.

Input: $(1,\ -{\frac {1}{2}})$
Desired Output: (1, 0, 0)
	Actual Output	Error Energy
Inverse Filter	(1, 0, −0.25)	0.063
Least-Squares Filter	(0.95, −0.09, −0.19)	0.048

Table 2-9. Design and application of a two-term least-squares inverse filter (a, b).

Filter Design
Convolution of the filter (a, b) with input wavelet $(-{\frac {1}{2}},\ 1)$:
	-  1/2	1		Actual Output	Desired Output
b	a			−a/2	1
	b	a		−b/2 + a	0
		b	a	b	0
Filter Application
Least-Squares Filter				(−0.95, −0.19)
Input Wavelet				(−0.5, 1)
Actual Output				(0.24, −0.38, −0.19)
Desired Output				(1, 0, 0)

Table 2-10. Error in two-term inverse and least-squares filtering.

Input: $(-{\frac {1}{2}},\ 1)$
Desired Output: (1, 0, 0)
	Actual Output	Error Energy
Inverse Filter	(1, 0, −4)	16
Least-Squares Filter	(0.24, −0.38, −0.19)	0.792

By simplifying equation (17), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we obtain

${\frac {5}{2}}a-b=-1,$

(18a)

and

$-a+{\frac {5}{2}}b=0.$

(18b)

Combine equations (18a,18b) into a matrix form

${\begin{pmatrix}5/2&-1\\-1&5/2\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}-1\\0\end{pmatrix}}.$

(19)

By solving for the filter coefficients, we obtain (a, b): (−0.95, −0.19). The design and application of this filter are summarized in Table 2-9.

Table 2-10 shows the results from the inverse filter and least-squares filter quantified. The error made by the least-squares filter is, again, much less than the error made by the truncated inverse filter. However, both filters yield larger errors for input wavelet (-  1/2, 1) (Table 2-10) as compared to errors for wavelet (1, -  1/2) (Table 2-8). The reason for this is discussed next.

See also

• The inverse of the source wavelet
• Minimum phase

External links

find literature about Least-squares inverse filtering